Let \(E\)  denote an elliptic curve over \(\mathbb {Q}\)  without complex multiplication with conductor \(N_E\) . For a prime \(p\)  of good reduction, \(E\)  reduces to an elliptic curve over the finite field \(\mathbb {F}_p\)  and we denote by \(a_p(E)\)  the trace of the Frobenius automorphism acting on the points of \(E\)  over \({\overline{\mathbb {F}}}_p\) . Then, \(a_p(E) = p+1-\# E(\mathbb {F}_p)\) , and the Hasse bound \(|a_p(E) | \le 2 \sqrt{p}\)  holds. The last few decades have seen much research pertaining to the distribution of the sequence \(a_p(E)/2\sqrt{p}\) . Sato and Tate formulated a conjecture for the distribution of this sequence in the interval \([-1,1]\)  around 1960 which was proven by Taylor, Clozel, Harris and Shepherd-Barron in [1]}, [2]}, [3]}.
